We investigate the computational complexity of the graph primality testing problem with respect to the direct product (also known as Kronecker, cardinal or tensor product). In [1] Imrich proves that both primality testing and a unique prime factorization can be determined in polynomial time for (finite) connected and nonbipartite graphs. The author states as an open problem how results on the direct product of nonbipartite, connected graphs extend to bipartite connected graphs and to disconnected ones. In this paper we partially answer this question by proving that the graph isomorphism problem is polynomial-time many-one reducible to the graph compositeness testing problem (the complement of the graph primality testing problem). As a consequence of this result, we prove that the graph isomorphism problem is polynomial-time Turing reducible to the primality testing problem. Our results show that connectedness plays a crucial role in determining the computational complexity of the graph primality testing problem.

我们针对直接乘积（也称为Kronecker，基数或张量积）调查图素测试问题的计算复杂性。在[1]中，Imrich证明，对于（有限）连通图和非二分图，可以在多项式时间内确定素数测试和唯一素数分解。作者提出了一个未解决的问题，即非二分连通图的直接积的结果如何扩展到二分连通图和不连续的图。在本文中，我们通过证明图同构问题是多项式时间多对一图可简化图综合性测试问题（图素性测试问题的补充）来部分回答该问题。结果，我们证明图同构问题是多项式时间图灵可归结为素性检验问题。我们的结果表明，连通性在确定图素测试问题的计算复杂度方面起着至关重要的作用。